Programming Assignment 1

Overview

In this assignment, you are to implement the Davis-Putnam procedure
and use it to solve the map-coloring problem.

Specifically: You will write three programs.

1. An implementation of the Davis-Putnam procedure, which takes
as input a set of clauses and outputs either a satisfying valuation,
or a statement that the clauses cannot be satisfied.

2. A front end, which takes as input a map to be colored and
a set of colors and outputs a set of clauses that can be input to (1).

3. A back end, which takes as input the output of the (1) and
translates it into a coloring of the map.

Specifications

Input / Output.

You have two choices as to how to organize your I/O:

1. All three programs take their input from standard input and
produce their output in standard output.

2. The front end reads from a file called "MAP.DAT" and writes to
a file "DP.DAT". The Davis-Putnam module reads from the file
"DP.DAT" and writes to a file called "DPSOL.DAT". The back end reads
from the file "DPSOL.DAT" and writes to a file called "COLORING.DAT".

Davis-Putnam

The input to the Davis-Putnam procedure has the following form:
An atom is denoted by a natural number: 1,2,3 ... The literal P
is the same number as atom P; the literal not P is the negative.
A clause is a line of text containing the integes of the corresponding
literals. After all the clauses have been given, the next line is the
single value 0; anything further in the file is ignored in the
execution of the procedure and reproduced at the end of the output file.
(This is the mechanism we will use to allow the front end to communicate
to the back end.)

The output from the Davis-Putnam procedure has the following form:
First, a list of pairs of atom (a natural number) and truth value
(either T or F). Second, a line containing the single value 0. Third,
the back matter from the input file, reproduced.

Example: Given the input

1 2 3
-2 3
-3
0
This is a simple example with 3 clauses and 3 atoms.

Davis-Putnam will generate the output

1 T
2 F
3 F
0
This is a simple example with 3 clauses and 3 atoms.

This corresponds to the clauses

P or Q or R.
not Q or R.
not R.

If the clauses have no solution, then Davis-Putnam outputs a single
line containing a 0, followed by the back-matter in the input file.

Note: Your implementation of Davis-Putnam must work on _any_ set of clauses,
not just those that are generated by the map coloring program

You may assume that there are no more than 1000 atoms and no more than 10,000
clauses.

Front end

The front end takes as input a representation of a map and generates as
output a set of clauses to be satisfied.

The format of the input contains the following elements:

A list of the colors, in a single line. A color is a character string
of up to 20 characters. The colors are separated by white space.

A list of
the names of the countries. These are character strings of up to 20
characters, separated by white space. The list of countries is terminated
by an asterisk.

A list of pairs of countries that border one another.
There is one pair per line.

Thus, the problem of coloring the map of New England + New York in three
colors is expressed in the following input file.

1. A set of clauses suitable for inputting to Davis-Putnam. As
explained in class, the clauses that you need for this problem are

A. For each country, a clause saying that the country has one of the
colors.

B. For each country, and for each pair of colors, a clause saying that
the country does not have both colors. (I.e. each country has a unique color).

C. For each pair of countries A,B that border one another and for each
color X, a clause stating that A and B are not both colored X.

Note that you can directly output the constraints in
clausal form (CNF) and therefore you do not have to implement
a program to translate arbitrary Boolean formulas to CNF.

2. A key for translating the numbers used for propositional atoms in
the clauses into pairs of countries and colors. This is to allow the
front end to communicate to the back end. The format of this is up to you.
My suggestion would be a sequence of lines of the form "number, country,
color".

Click below for the files illustrating the steps in one possible solution
of the problem:

You may assume that there are no more than 10 colors and 100 countries.

Back-end

The back end takes as input the output that Davis-Putnam generates when run
on the output of the front end. It generates as output a coloring of
the map. The format is pairs of the form "country color", one pair per line.
For example, one possible output for the above example would be

Programming Language

My personal preference would be for Java or C; however, if you like something
else that's also OK. Keep in mind that the onus is on you to demonstrate
(a) that the program works (b) that it implements the Davis-Putnam algorithm
correctly. As regards (a), if your program works fine on your machine and bombs
when I try it, it's your responsibility, not mine, to fix the problem. So
portability should be a priority, which is why I suggest Java.
As regards (b), I will judge that by reading through your code, so you
should try to write elegant, readable, well commented code.

Deliverable

You should email me (davise@cs.nyu.edu)
(a) the source code; (b) instructions for running it,
if there's anything at all non-obvious about it. Nothing else.

Grading

The Davis-Putnam program is worth 60% of the grade; the front end is worth
30%; the back end is worth 10%. In each of these, a program that does
not compile will get a maximum of 10%; a correct program
will get 90%; the remaining 10% is for being well-written and
well commented.